iSoul Time has three dimensions

Coordinate transformations

Coordinate Transformations with t

x = space coordinate, t = time coordinate, v = velocity, u = pace

Galilean transformation

speed: = xvt and t´ = t,         pace: = xt/u and t´ = t.

Dual Galilean transformation

speed: = tx/v and x´ = x,       pace: = tux and x´ = x.

Light:   c := speed of light,        := pace of light.

speed: x = ct or x/c = t and = ct´, or x´/c = ,

pace:  c´x = t or x = t/c´ and c´r′ = or = t´/c´.

Lorentz transformation

speed: γ = (1 – v2/c2)–1/2 with x´ = γ (x − vt) and = γ (txv/c²),

pace:  γ = (1 – 2/u2)–1/2 with γ (xt/u) and t´γ (txc´²/u),

which applies only if |v| < |c| or |u| > ||.

Dual Lorentz transformation

speed: λ = (1 − c2/v2)–1/2 with λ (t − x/v) and λ (xt (c2/v)),

pace:  λ = (1 – u2/2)–1/2 with λ (tux) and λ (xt (u/c´²)),

which applies only if |v| > |c| or |u| < ||.

Galileo-Lorentz limit

speed: c1 → ∞: γ1 → 1, t´t,                c2c/2: γ2 → (1 – 4v²/c²)−1/2 and t´γ2 (t – 4x (v/c²)).

pace:  1 → 0: γ1 → 1, t´t,                 ç2 → 2/: γ2 → (1 – (1/4) ²/u²)−1/2 and t′ → γ2 (t – (x/4) (²/u)).

Dual Galileo-Lorentz limit

speed: c1 → 0: λ1 → 1, x´x,               c2 → 2/c: λ2 → (1 – (1/4) c²/v²)−1/2 and x´λ2 (x – (t/4) (c²/v)).

pace:  1 → ∞: λ1 → 1, x´x,               2/2: λ2 → (1 – 4u²/²)−1/2 and x´λ2 (x – 4t (u/²)).

If |v| = |c|, then x′ = x and t′ = t.

Coordinate Transformations with τ

x = space coordinate, t = time coordinate, v = velocity, u = pace

c := speed of light,        := pace of light.

τ := ct                           τ := t/

Galilean transformation

speed: = xτ (v/c) and τ´ = τ,              pace: = xτ (c´/u) and τ´ = τ.

Dual Galilean transformation

speed: τ´ = τx (c/v) and x´ = x,              pace: τ´ = τx (u/) and x´ = x.

Light: x = τ and = τ´

Lorentz transformation

speed: γ = (1 – v2/c2)–1/2 with = γ (x − τ (v/c)) and τ´ = γ (τx (v/c)),

pace:  γ = (1 – 2/u2)–1/2 with γ (xτ (/u)) and τ´γ (τx (/u)),

which applies only if |v| < |c| or |u| > ||.

Dual Lorentz transformation

speed: λ = (1 − c2/v2)–1/2 with τ′λ (τ – x (c/v)) and λ (xτ (c/v)),

pace:  λ = (1 – u2/2)–1/2 with τ′λ (τx (u/)) and λ (xτ (u/)),

which applies only if |v| > |c| or |u| < ||.

Galileo-Lorentz limit

speed: c1 → ∞: γ1 → 1, τ´τ,               c2c/2: γ2 → (1 – 4v²/c²)−1/2 and τ′ → γ2 (τ – 4x (v/c)).

pace:  ç1 → 0: γ1 → 1, τ´τ,                 ç2 → 2/: γ2 → (1 – (1/4) ²/u²)−1/2 and τ′ → γ2 (τ – (x/4) (/u)).

Dual Galileo-Lorentz limit

speed: c1 → 0: λ1 → 1, x´x,               c2 → 2/c: λ2 → (1 – (1/4) c²/v²)−1/2 and x´λ2 (x – (τ/4) (c/v)).

pace:  ç1 → ∞: λ1 → 1, x´x,               ç2/2: λ2 → (1 – 4u²/²)−1/2 and x´λ2 (x – 4τ (u/)).

If |v| = |c|, then x′ = x and τ′ = τ.

Coordinate Transformations with c := 1

x = space coordinate, t = time coordinate, v = velocity, u = pace

Galilean transformation

speed: = xvt and t´ = t,         pace: = xt/u and t´ = t.

Dual Galilean transformation

speed: = tx/v and x´ = x,       pace: = tux and x´ = x.

Light:   1 = c := speed of light,  1 =  := pace of light.

speed: x = t and x´ =

pace:  x = t and = .

Lorentz transformation

speed: γ = (1 – v2)–1/2 with = γ (x − vt) and = γ (txv),

pace:  γ = (1 – 1/u2)–1/2 with γ (xt/u) and γ (tx/u),

which applies only if |v| < 1 or |u| > 1.

Dual Lorentz transformation

speed: λ = (1 − 1/v2)–1/2 with λ (t − x/v) and λ (xt (1/v)),

pace:  λ = (1 – u2)–1/2 with t′λ (tux) and λ (xtu),

which applies only if |v| > 1 or |u| < 1.

If |v| = 1, then x´ = x and t´ = t.